For a polynomial $p(z)$ of degree $n$ which has no zeros in$|z|< 1$, Dewan et al., (K. K. Dewan and Sunil Hans, Generalizationof certain well known polynomial inequalities, J. Math. Anal.Appl., 363 (2010), 38--41) established$$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big|\leq\frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$for any complex number $\beta$ with $|\beta|\leq 1$ and $|z|=1$.In this paper we consider the operator $B$, which carries apolynomial $p(z)$ into$$B[p(z)]:=\lambda_0p(z)+\lambda_1\Big(\frac{nz}{2}\Big)\frac{p'(z)}{1!}+\lambda_2\Big(\frac{nz}{2}\Big)^2\frac{p''(z)}{2!}, $$ where $\lambda_0,$ $\lambda_1$, and $\lambda_2$ are such that all the zeros of $u(z)=\lambda_0+c(n,1)\lambda_1z+c(n,2)\lambda_2z^2$ lie in the half plane $|z|\leq |z-{n}/{2}|.$By using the operator $B$, we present a generalization of resultof Dewan. Our result generalizes certain well-known polynomialinequalities.